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08. Bernoulli Equations

Dated: 10-11-2024

Any differential equation of the form

\[\frac{dy}{dx}+p(x)y=q(x)y^n\]

is called Bernoulli equation.

Method of Solution

  • For \(n = 0, 1\), the equation reduces to first order linear differential equation.1
  • For \(n \ne 0, 1\) we divide by \(y^n\)
\[y^{-n}\frac{dy}{dx}+p(x)y^{1-n}=q(x)\]
  • Substitute
\[v=y^{1-n}\]
  • Differentiate2 with respect to \(x\).
\[v^\prime=(1-n)y^{-n}y^\prime\]
\[\frac{dv}{dx}+(1-n)p(x)v=(1-n)q(x)\]

After solving, we will get

\[y=v^{\frac{1}{1-n}}\]

if \(n > 0\) then we will add \(y = 0\) to the solution.

Example

\[\frac{dy}{dx}=y+y^3 \quad p(x) = -1, q(x) = 1, n = 3\]

Step 1

\[\frac{dy}{dx}-y=y^3\]

Step 2

Dividing by \(y^3\)

\[y^{-3}\frac{dy}{dx}-y^{-2}=1\]

Step 3

\[v=y^{1-3}=y^{-2}\]

Step 4

\[y^{-3}\frac{dy}{dx}=-\frac{1}{2}\left(\frac{dv}{dx}\right)\]
\[\frac{dv}{dx}+2v=-2\]

Step 5

\[u(x)=e^{\int2dx}=e^{2x}\]

Step 6

\[v=\frac{\int u(x)q(x)dx+c}{u(x)}=\frac{\int e^{2x}(-2)dx+c}{e^{2x}}\]
\[\because \int e^{2x}(-2)dx=-e^{2x}\]
\[v=\frac{-e^{2x}+C}{e^{2x}}=Ce^{-2x}-1\]

Step 7

\[y=\pm(Ce^{-2x}-1)^{-\frac{1}{2}}\]

Therefore the solutions are

\[ \begin{cases} y=0 \\ y=\pm(Ce^{-2x}-1)^{-\frac{1}{2}} \end{cases} \]

Example

\[y(1+2xy)dx+x(1-2xy)dy=0\]

This equation is not - Separable3 - Homogeneous4 - Exact5 - Linear1 - Bernoulli6

After staring long enough, we notice that

\[u = 2xy\]
\[y = \frac u {2x}\]
\[\because dy = \frac{xdu - udx}{2x^2}\]
\[2u^2dx+(1-u)xdu=0\]
\[2\ln|x|-u^{-1}-\ln|u|=c\]
\[\ln|\frac{x}{2y}|=c+\frac{1}{2xy}\]
\[\frac{x}{2y}=c_{1}e^{\frac 1 {2xy}}\]
\[x=2c_{1}ye^{\frac 1 {2xy}}\]

Example

\[\frac{d^{2}y}{dx^{2}}=2x\left(\frac{dy}{dx}\right)^{2}\]
\[u = y^\prime\]
\[\frac{du}{dx}=y^{\prime\prime}\]
\[\frac{du}{dx}=2xu^2\]
\[\frac{du}{u^2}=2xdx\]
\[\int u^{-2}du=\int 2xdx\]
\[-u^{-1}=x^2+c_1\]
\[u^{-1}=\frac{1}{y'}\]
\[\frac{dy}{dx}=-\frac{1}{x^{2}+c_{1}^{2}}\]
\[dy=-\frac{dx}{x^{2}+c_{1}^{2}}\]
\[\int dy=-\int \frac{dx}{x^{2}+c_{1}^{2}}\]
\[y+c_{2}=-\frac{1}{c_{1}}\tan^{-1}\frac{x}{c_{1}}\]

References

Read more about notations and symbols.

  1. Read more about first order differential equation. ↩↩

  2. Read more about differentiation. ↩

  3. Read more about separable equations. ↩

  4. Read more about homogeneous differential equations. ↩

  5. Read more about exact differential equations. ↩

  6. Read more about bernoulli equation. ↩